Numbers, units and physical quantities.

Okay, it may be similar to teaching your grandmother to… but it’s important to remember, and you really shouldn’t skip anything, no matter how trivial it seems.

Scientific Notation

A non-SI unit of pressure is 1atm (one atmosphere), which when expressed in SI units is 101,325 pascal. The charge on an electron in SI units is -0.0000000000000000001602 coulomb. You can see the problem, huge range of values to deal with. The best way is to use Scientific Notation. 1atm then becomes 1.01325 X 105 pascal, and the charge on the electron (e) = 1.602 X 10-19 coulomb.

In Scientific notation any number can be written as the product of a number r such that 1 ≤ |r| < 10

When a number is describing a physical quantity, each of the figures should have a meaning, each should be a significant figure. Stating that a length is 6.00mm should mean that it really is 6.00mm and not 6.01mm or 5.99mm. The use of three significant figures in this example implies that the measurement is known to the nearest 0.01mm i.e. between 5.999mm and 6.005mm. The number of significant figures in a measurement should indicate the precision with which that value is known.

The Significant figures in a number are the meaningful digits that indicate its precision. They do not include any zeros to the left of the first non-zero digit. Using scientific notation avoids the need to write down any zeros that are not significant, either to the left or to the right of the significant figures.

When multiplying or dividing two numbers, the result should not be quoted to more significant figures than the least precisely determined number e.g. 0.4 X 1.21 = 0.5 (not 0.484 as my calculator seems to think!)

When adding or subtracting two numbers, the last significant figure in the result should be the last significant figure that appears in both the numbers when expressed in decimal form without powers of ten, e.g. 0.4 + 1.21 = 1.6 and 0.004 + 1.21 = 1.21. That last one might look odd but nevertheless true!

When calculations are performed with a consistent number of significant figures throughout, the final result may still not be reliable to that number of significant figures.

Natural Numbers and Integers.

For the purpose of counting only the positive whole numbers are needed, these are called the Natural Numbers. They form the set ℕ i.e ℕ = {1,2,3…}

Each object (number) belonging to a set is referred to as an element e.g. 57 ∈ ℕ

Another set of numbers are the Integers ℤ, these are the natural numbers, and zero and the negative numbers, ℤ = {… -2, -1, 0, 1, 2 …}

The whole set of ℕ is included within the set ℤ, i.e ℕ is a subset of ℤ , ℕ ⊂ ℤ

All the natural and integers numbers are included in the set of Real Numbers ℝ which is the set of all positive and negative decimal numbers e.g. 4.01 ∈ ℝ

The set of real numbers also contains the set of Rational ℚ and Irrational numbers (there is no symbol 🙂 ). Rational numbers are ones that can be expressed as a fraction (or quotient, hence the Q) e.g ½. Irrational numbers such as π or √2 cannot be expressed precisely as fractions (ratio of two integers).